g#dZddlmZddlmZmZmZmZmZm Z m Z ddl m Z ddl mZmZdZdZdZifd Zd Zd Zd Zd ZdZdZdS)a&Implementation of DPLL algorithm Further improvements: eliminate calls to pl_true, implement branching rules, efficient unit propagation. References: - https://en.wikipedia.org/wiki/DPLL_algorithm - https://www.researchgate.net/publication/242384772_Implementations_of_the_DPLL_Algorithm )default_sort_key)OrNot conjuncts disjunctsto_cnf to_int_repr_find_predicates)CNF)pl_trueliteral_symbolct|tstt|}n|j}d|vrdSt t |t}ttdt|dz}t||}t||i}|s|Si}|D](}| ||dz ||i)|S)a> Check satisfiability of a propositional sentence. It returns a model rather than True when it succeeds >>> from sympy.abc import A, B >>> from sympy.logic.algorithms.dpll import dpll_satisfiable >>> dpll_satisfiable(A & ~B) {A: True, B: False} >>> dpll_satisfiable(A & ~A) False F)key) isinstancer rrclausessortedr rsetrangelenr dpll_int_reprupdate)exprrsymbolssymbols_int_reprclauses_int_reprresultoutputrs W/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/logic/algorithms/dpll.pydpll_satisfiabler s dC F4LL)), u%d++1ABBBG5CLL1$45566"7G44 +-=r B BF  F77 wsQw'56666 Mct||\}}|rV|||i|||s|}t||}t||\}}|Vt ||\}}|rV|||i|||s|}t||}t ||\}}|Vg}|D]2}t ||}|durdS|dur||3|s|S|s|S|}|}||di||di|dd} tt||||p+tt|t|| |S)z Compute satisfiability in a partial model. Clauses is an array of conjuncts. >>> from sympy.abc import A, B, D >>> from sympy.logic.algorithms.dpll import dpll >>> dpll([A, B, D], [A, B], {D: False}) False FTN) find_unit_clauserremoveunit_propagatefind_pure_symbolr appendpopcopydpllr rrmodelPvalueunknown_clausescval model_copy symbols_copys rr*r*1s //HAu 4 aZ   q A !,,#GU335 4 11HAu 6 aZ   q A !,,#GW555 6O &&a %<<55 d??  " "1 % % %    AJ LL!Tq%j!!!111:L 33We D D T Q88, S SUr!cvt||\}}|rV|||i|||s| }t||}t||\}}|Vt ||\}}|rV|||i|||s| }t||}t ||\}}|Vg}|D]2}t ||}|durdS|dur||3|s|S|}|}||di||di|} tt||||ptt|| | |S)z Compute satisfiability in a partial model. Arguments are expected to be in integer representation >>> from sympy.logic.algorithms.dpll import dpll_int_repr >>> dpll_int_repr([{1}, {2}, {3}], {1, 2}, {3: False}) False FT) find_unit_clause_int_reprrr$unit_propagate_int_reprfind_pure_symbol_int_reprpl_true_int_reprr'r(r)rr+s rrrbs)%88HAu = aZ   q A)'155,We<<5 =)'::HAu ? aZ   q A)'155,Wg>>5 ?O &&q%(( %<<55 d??  " "1 % % %   AJ LL!Tq%j!!!<<>>L 1/1EEwPU V V b 1/A2FF V` a acr!cd}|D]D}|dkr|| }|| }n||}|durdS|d}E|S)af Lightweight version of pl_true. Argument clause represents the set of args of an Or clause. This is used inside dpll_int_repr, it is not meant to be used directly. >>> from sympy.logic.algorithms.dpll import pl_true_int_repr >>> pl_true_int_repr({1, 2}, {1: False}) >>> pl_true_int_repr({1, 2}, {1: False, 2: False}) False FrNT)get)clauser,rlitps rr8r8snF   77 3$A}E #A 9944 YF Mr!cg}|D]}|jtkr||(|jD]@}|kr/|tfd|jDn|krnA|||S)a Returns an equivalent set of clauses If a set of clauses contains the unit clause l, the other clauses are simplified by the application of the two following rules: 1. every clause containing l is removed 2. in every clause that contains ~l this literal is deleted Arguments are expected to be in CNF. >>> from sympy.abc import A, B, D >>> from sympy.logic.algorithms.dpll import unit_propagate >>> unit_propagate([A | B, D | ~B, B], B) [D, B] c"g|] }|k | Sr@).0xsymbols r z"unit_propagate..s "E"E"EfW 1 r!)funcrr'args)rrCrr0args ` rr%r%s"F    6R<< MM!    6  Cvg~~ b"E"E"E"Eaf"E"E"EFGGGf}} MM!    Mr!c, hfd|DS)z Same as unit_propagate, but arguments are expected to be in integer representation >>> from sympy.logic.algorithms.dpll import unit_propagate_int_repr >>> unit_propagate_int_repr([{1, 2}, {3, -2}, {2}], 2) [{3}] c"g|] }|v|z  Sr@r@)rAr;negatedss rrDz+unit_propagate_int_repr..s# F F FavooFW ooor!r@)rrKrJs `@rr6r6s,rdG F F F F F7 F F FFr!c|D]O}d\}}|D]9}|s|t|vrd}|s t|t|vrd}:||kr||fcSPdS)a# Find a symbol and its value if it appears only as a positive literal (or only as a negative) in clauses. >>> from sympy.abc import A, B, D >>> from sympy.logic.algorithms.dpll import find_pure_symbol >>> find_pure_symbol([A, B, D], [A|~B,~B|~D,D|A]) (A, True) )FFTNN)rr)rr/sym found_pos found_negr0s rr&r&s""+ 9  ! !A ! ! !4!4  !SYq\\!9!9  ! ! > ! ! ! " :r!ctj|}||}|d|D}|D] }| |vr|dfcS|D]}| |vr| dfcSdS)a Same as find_pure_symbol, but arguments are expected to be in integer representation >>> from sympy.logic.algorithms.dpll import find_pure_symbol_int_repr >>> find_pure_symbol_int_repr({1,2,3}, ... [{1, -2}, {-2, -3}, {3, 1}]) (1, True) cg|]}| Sr@r@)rArKs rrDz-find_pure_symbol_int_repr..s)>)>)>1")>)>)>r!TFrM)runion intersection)rr/ all_symbolsrOrPr=s rr7r7s#%%+/K((11I(()>)>g)>)>)>??I  2Y  d7NNN   2Y  2u9     :r!c|D]R}d}t|D]2}t|}||vr|dz }|t|t }}3|dkr||fcSSdS)a A unit clause has only 1 variable that is not bound in the model. >>> from sympy.abc import A, B, D >>> from sympy.logic.algorithms.dpll import find_unit_clause >>> find_unit_clause([A | B | D, B | ~D, A | ~B], {A:True}) (B, False) rrrM)rr rr)rr,r;num_not_in_modelliteralrNr-r.s rr#r# s (( = =G ))C% A% Jw$<$< <5 q e8OOO ! :r!ct|d|Dz}|D]A}||z }t|dkr'|}|dkr| dfcS|dfcSBdS)a Same as find_unit_clause, but arguments are expected to be in integer representation. >>> from sympy.logic.algorithms.dpll import find_unit_clause_int_repr >>> find_unit_clause_int_repr([{1, 2, 3}, ... {2, -3}, {1, -2}], {1: True}) (2, False) ch|]}| Sr@r@)rArNs r z,find_unit_clause_int_repr..+s00033$000r!rrFTrM)rrr()rr,boundr;unboundr=s rr5r5 s JJ00%000 0E5. w<<1   A1uur5y   $w  :r!N)__doc__sympy.core.sortingrsympy.logic.boolalgrrrrrr r sympy.assumptions.cnfr sympy.logic.inferencer r r r*rr8r%r6r&r7r#r5r@r!rrcs]0/////""""""""""""""""""%%%%%%99999999>.U.U.Ub+c+c+c`$&6B G G G..,r!